inharmonicity

Inharmonicity refers to the deviation of the overtones (partials) of a vibrating system from exact integer multiples of the fundamental frequency. In an ideal, perfectly flexible string or air column, the harmonic series would have overtones at frequencies f_n = n f_1. Real instruments, however, exhibit inharmonicity when stiffness, geometry, and boundary conditions cause the higher partials to lie above this ideal series.

Common causes include the stiffness of strings and bars, the geometry of bells or membranes, and nonuniform material properties. The degree of deviation is often summarized by an inharmonicity coefficient: larger coefficients indicate stronger deviations. In stiff strings, the effect tends to increase with the square of the overtone number, making higher partials noticeably sharper than exact multiples.

Inharmonicity shapes timbre and has practical implications for tuning and instrument design. On the piano, for example, the upward shift of higher partials complicates equal temperament tuning, leading to a practice known as stretch tuning, where higher notes are tuned slightly sharper to preserve consonance across the scale. Other instruments, including bells, xylophones, and certain membranes, exhibit pronounced inharmonic spectra that contribute to their distinctive sounds.

Measurement of inharmonicity involves comparing actual partial frequencies to the ideal harmonic series, commonly by examining the ratios f_n/(n f_1). This metric helps characterize a instrument’s spectral structure and informs decisions in performance, construction, and acoustical analysis. See also: harmonic series, timbre, piano tuning, string instrument.